The 3rd order Weighted Essentially Non-Oscillatory (WENO) scheme and a recently developed Dynamic Adaptive Chemistry (DAC) method are integrated with the hybrid multi-timescale (HMTS) method and an adaptive unstructured grid system for computationally efficient and adaptive direct numerical simulation. The results show that compared to the 1st order total variation diminishing (TVD) scheme, the 3rd order WENO scheme dramatically reduces the numerical dissipation while keeping good computational efficiency. In the dynamic adaptive chemistry method, the kinetic mechanism is dynamically reduced by the Path Flux Analysis (PFA) method on each computational grid. A hybrid multi-timescale method, which combines the implicit and explicit Euler schemes, is used to solve the chemical reactions based on the dynamically reduced mechanism. The algorithm is validated and applied for simulation of ignition and unsteady flame propagation of n-decane/1,2,4 trimethyl benzene (TMB) diesel fuel surrogate mixtures. The results show the present algorithm is not only computationally efficient but also robust and accurate. The DAC method increases the computation efficiency dramatically. Moreover, the unsteady flame propagation simulation shows that the 3rd order WENO scheme can significantly reduce the numerical dissipation and predict a more accurate flame speed compared to the 1st order scheme.